Stress formulation and duality approach in periodic homogenization

TL;DR

This paper explores various formulations of the cellular problem in periodic homogenization, including primal and dual variational approaches, to facilitate numerical solutions in material modeling.

Contribution

It introduces multiple variational and dual formulations of the cellular problem, enhancing the mathematical and computational tools for homogenization analysis.

Findings

Multiple variational formulations presented

Dual formulations in displacement-stress and strain-stress derived

Lagrangians suitable for numerical optimization algorithms

Abstract

This paper describes several different formulations of the so-called "cellular problem" which is a system of partial differential equations arising in the theory of homogenization, subject to periodicity boundary conditions. Variational formulations of the cellular problem are presented where the main unknown is the displacement, the stress or the strain, as well as several different formulations as minimization problems. Two dual formulations are also presented, one in displacement-stress and another one in strain-stress. The corresponding Lagrangians may be used in numerical optimization algorithms based on alternated directions.